Mathematical programming and Simplex method
Mathematical programming techniques can be applied to problems
where one wants to minimize (or maximize) a function, called the objective
function, subject to a set of constraints. A solution is
said to be feasible if it satisfies
the constraints. A solution is said to be optimal
if it is feasible and yields the smallest (or largest) value of the objective
function among all feasible solutions. A problem may have more than one
solution that yields the optimal value.
In linear programming, both the objective function and the constraints
are linear and so a linear programming (LP) problem can be formulated in
the following form:
lb(i) <= A(i,*)x <= ub(i) for i = 1,..., m
lc(j) <= x(j) <= uc(j) for j = 1,..., n
Another way of defining a linear programming problem is the following:
A(i,*)x sgn b(i) for i = 1,..., m
lc(j) <= x(j) <= uc(j) for j = 1,..., n
To deal with the problem in a form that is easier to manipulate, logical
variables can be added to create equalities and variable substitution can
be performed to eliminate negative variables. The result is a more textbook
Ax = b
x >= 0
Geometrically, the constraints can be thought of as defining an (n -
m)-dimensional polyhedron, in an n-dimensional space.This is
known as the feasible region.
The goal is to find a point on the polyhedron that minimizes the objective
function. This is called the optimal solution. If such a solution exists,
the problem is feasible; otherwise, the problem is infeasible
The above problem is known as the primal
problem. There is a related problem called the dual
of the primal problem. The dual problem shown below provides
a lower bound for the primal problem. Also, the optimal value for the primal
problem is equal to the optimal value of the dual problem.
y'A <= c
For solving linear programming problems, there are several approaches
which are discussed below.
Linear Programming: Simplex Method
The simplex method for solving linear programming problems was developed
by Dantzig in 1947. It is an iterative
procedure for finding an optimal solution. In 1975 the Nobel prize in economics
has been awarded to the famous mathematician L. V. Kantorovich
for his contribution in the application of linear programming to the
solution of problem of allocation of scarce resources.
It can be shown that if an optimum exists, there is an optimal solution
among the vertices of the polyhedron defining the feasible region. Any
solution that is a vertex of the polyhedron defining the feasible region
is called a basic feasible solution (BFS).
The primal simplex method searches for an optimal solution among the
BFSs. A BFS corresponds to using m columns of the coefficient
matrix A as a basis for the vector space spanned by the columns
of A. Given an initial BFS, a new BFS is computed by choosing one
of the remaining (n - m) columns to enter the basis and one of
the current m basic columns to leave the basis. The entering column
is chosen so the value of the primal objective function decreases, or at
least does not increase, when this column enters the basis; the exiting
column is chosen so that the new BFS is still primal feasible.
In the primal simplex algorithm, you begin with a primal feasible solution
which may be dual infeasible. At each step the primal objective function
is reduced by introducing variables into the basis corresponding to infeasibilities
in the dual problem. An optimum BFS is detected when there are no remaining
In the dual simplex algorithm, you begin with a dual feasible solution
which may be primal infeasible. At each step, the dual objective function
is increased by removing variables from the basis corresponding to infeasibilities
in the primal problem. An optimum BFS is detected when there are no remaining
Linear Programming: Interior-Point Barrier Method
Interior-point methods for linear programming problems have only been shown
to be computationally efficient in recent years. Interior-point methods
are also iterative techniques for finding an optimal solution.
While the simplex algorithm searches for the optimal solution among
the vertices of the feasible polyhedron, interior-point methods search
for solutions in the interior of the feasible region. OSL contains three
interior-point methods that are types of the logarithmic barrier method.
These are the primal barrier method, the primal-dual barrier method, and
the primal-dual method with the predictor-corrector feature. The latter
will often be referred to simply as the predictor-corrector method. The
primal barrier method takes a primal linear programming problem and solves
a sequence of subproblems of the form:
'minimize' F(x;m) = c'x - m* S ln x(j)
Ax = b
mu > 0.
Here c, x, A, and b are as defined above. The scalar m
is known as the barrier parameter
and is specified for each subproblem. The equality constraints are handled
directly. Then each function F(x;m)>
is approximately optimized to obtain a solution x(m).
If x* is an optimal solution to the primal linear programming problem,
then the sequence of approximate solutions x(m)
converge to x* as m
converges to 0.
Given an initial feasible interior solution, a new solution is computed
by choosing a direction to travel from the initial point and a distance
to travel along that direction. The direction and distance are chosen so
that the value of the objective function decreases along the step and so
that the step taken is relatively long. Primal-dual barrier methods consider
not only the primal problem but the dual as well. The dual problem can
be written as:
A'y + z = c
z >= 0
Using a barrier parameter m,
this problem may be considered as a sequence of subproblems of the form:
'maximize' b'y + m * S ln z(i)
A'y + z = c
m > 0
Assuming the same sequence of m
values for the primal and dual problems, the necessary conditions for optimality
of each subproblem are:
Ax = b
A'y + z = c
XZ e = m e
where e is a vector of 1's and X and Z are
diagonal matrices with the elements of x and z, respectively,
as the diagonal elements. Note that this product of variables makes the
subproblem nonlinear. It may be solved (or partially solved) by the classical
approach of Newton's method for each value of m
considered. In practice, only one Newton iteration is performed for each
value of m. When
x, y, and z are feasible, they approach primal and
dual optimal solutions as m
approaches zero. That the feasible primal and dual problems must have identical
optimal solution values provides a check-and-control mechanism for convergence.
Linear Programming: Sensitivity Analysis
Sensitivity analysis determines the effect that changes in objective
function coefficients (costs) or the row and column bounds (for example,
limits on availability) have on the solution. Suppose an oil company blends
gasoline using a few types of oil out of the many available. There may
be many choices of types of oil that give the desired results, but one
mixture optimizes the objective function. Solving the LP gives the optimal
answer, that is, how much of which types of oil should be used to give
gasoline meeting the requirements at the minimum cost.
After obtaining the optimal solution you may want to know how sensitive
the choices of oil types are to the coefficients in the objective function.
These coefficients might represent the current selling prices of gasoline
and the purchase prices of oil. You may also want to know how sensitive
the solution is to the upper and lower bounds on rows and columns. These
bounds may represent availability. Sensitivity analysis determines how
much these factors have to change to cause the optimal solution to occur
at a different set of activity values. This is essentially asking: What
changes would cause the solution to occur with a different basis? This
is what is meant by sensitivity analysis.
Linear Programming: Parametrics
Parametric analysis is similar to sensitivity analysis, but instead of
asking what changes in a single cost or bound are necessary to cause the
solution to occur at a different basis, a range of values is specified
for the objective function coefficients and bounds. OSL then determines
what happens to the solution as the objective function coefficients and
bounds are varied simultaneously through their specified ranges.
Again, think of the oil company that has an optimal solution for blending
gasoline from crude oils. Instead of just answering how much of a change
in the assumptions is required to cause a basis change, parametric analysis
gives the optimal for a entire range of assumptions. For example, assume
that when the oil company LP was solved, type 1 crude oil cost $20 per
barrel. Sensitivity analysis might say that the basis would change if the
price of type 1 crude oil rose to $23 per barrel or fell to $18 per barrel.
Parametric analysis, however, can answer the question: ``What is the optimal
solution for every price of type 1 crude oil between $10 and $20 in $2
increments?'' In addition, the parametric algorithm reports on all the
solutions at basis changes between $10 and $20.
It should be noted that parametric analysis usually requires the solution
of many LP problems, and so it may take considerably longer to run than
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